Hypercyclic Conjugate Operators

We prove that for any weighted backward shift B = B _w on an infinite dimensional separable Hilbert space H whose weight sequence w = (w _n ) satisfies \( \sup_{n} {\left| {w_{1} w_{2} \ldots w_{n} } \right|} = \infty \), the conjugate operator \( C_{B} :S \mapsto BSB^{*} \) is hypercyclic on the space S(H) of self-adjoint operators on H provided with the topology of uniform convergence on compact sets. That is, there exists an \( S \in S(H) \) such that \( \{ C^{n}_{B} (S) = B^{n} SB^{*n}\} _{{n \geq 0}} \) is dense in S(H). We generalize the result to more general conjugate maps \( S \mapsto TST^{*} \), and establish similar results for other operator classes in the algebra B(H) of bounded operators, such as the ideals K(H) and N(H) of compact and nuclear operators respectively.

Authors and Affiliations

1. School of Mathematical Sciences, Chalmers/Göteborg University, SE-412 96, Göteborg, Sweden

   Henrik Petersson

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Correspondence to Henrik Petersson.

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